Chapter 4: Clustering and classification
date()## [1] "Tue Nov 29 00:16:45 2022"Analysis exercises
Task 2.
The data set, titled “Boston”, used in this exercise contains 14 variables (columns) and 506 observations (rows). It entails information about housing values used to evaluate the willingness of people to pay for cleaner air in the Boston metropolitan area in the 1970s (source).
library(MASS)
data("Boston")
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...dim(Boston)## [1] 506 14Task 3.
Looking at the plots and summary table, it looks like most variables are not equally distributed, such as crim, zn, and rad. Also many of the variables have strong correlations (both positive and negative) with each other based on the plots. The weakest correlating variable seems to be chas, i.e. the Charles River dummy variable. This might be due to it being a binary variable (0 or 1). Interestingly, indus, i.e. the proportion of non-retail business acres per town, and tax, i.e. full-value property-tax rate per 10 000 $, are bimodally distributed.
library(GGally)
ggpairs(Boston, lower = list(
continuous = wrap(
"smooth",
alpha = 0.3,
size = 0.5,
color = "firebrick1"
)
)) + theme_bw()
library(tidyverse)
library(corrplot)
cor_matrix <- cor(Boston) %>% round(digits = 2)
corrplot(
cor_matrix,
method = "circle",
type = "lower",
cl.pos = "b",
tl.pos = "d",
tl.cex = 0.6
)
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00Task 4.
The original Boston data set had variables measured in different ways and scales/magnitudes. After using scale(), all variables and observations have been normalized to the same scale. This allows for better comparison between them.
boston_scaled <- Boston %>% scale()
summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865boston_scaled_df <- as.data.frame(boston_scaled)
bins <- quantile(boston_scaled_df$crim)
crime <-
cut(
boston_scaled_df$crim,
breaks = bins,
include.lowest = TRUE
)
boston_scaled_df <- boston_scaled_df %>% dplyr::select(-crim)
boston_scaled_df <- data.frame(boston_scaled_df, crime)
levels(boston_scaled_df$crime) <- c("low", "med_low", "med_high", "high")
n <- nrow(boston_scaled_df)
set.seed(2126)
ind <- sample(n, size = n * 0.8)
train <- boston_scaled_df[ind,]
test <- boston_scaled_df[-ind,]Task 5.
lda.fit <- lda(crime ~ ., data = train)
lda.arrows <-
function(x,
myscale = 1,
arrow_heads = 0.1,
color = "black",
tex = 0.75,
choices = c(1, 2)) {
heads <- coef(x)
arrows(
x0 = 0,
y0 = 0,
x1 = myscale * heads[, choices[1]],
y1 = myscale * heads[, choices[2]],
col = color,
length = arrow_heads
)
text(
myscale * heads[, choices],
labels = row.names(heads),
cex = tex,
col = color,
pos = 3
)
}
classes <- as.numeric(train$crime)
plot(lda.fit,
dimen = 2,
col = c("red", "blue", "purple", "gold")[classes])
lda.arrows(lda.fit, myscale = 2)
Task 6.
The table shows that the majority of the predicted results of our model overall are correct (74 out of 102, 72.55%). Looking at the table in more detail, the model predicted correctly 16 out of 25 (64.0%) low crime rates, 16 out of 28 (57.14%) medium low crime rates, 12 out of 18 (66.67%) medium high crime rates, and 30 out of 31 (96.77%) high crime rates. This shows that the model is highly accurate in predicting high crime rates, however is somewhat inaccurate with lower crime rates, especially with medium low ones.
correct_classes <- test$crime
new_test <- test %>% dplyr::select(-crime)
lda.pred <- predict(lda.fit, newdata = new_test)
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 16 5 1 0
## med_low 8 16 5 0
## med_high 1 7 12 1
## high 0 0 0 30Task 7.
As it was not specified which distances we were supposed to compute, might as well compute both Euclidian and Manhattan distances. For the initial k-means analysis, I randomly picked three clusters. But after determining the optimal number of clusters the re-analysis with the total of within cluster sum of squares, I ended up with two based on the line plot. That was due to the steepest drop being between one and two clusters. The scaling of the Boston dataset seems to do weird things to the distribution of the rad variable. I’m not sure why. Anyway, looking at the new distribution with k-means of two clusters, the indus, nox, and tax variable have two different colored peaks. This makes sense as I previously stated that these are bimodally distributed. I would assume that the scaled data has two centroids.
data("Boston")
new_boston_scaled <- as.data.frame(scale(Boston))
dist_eu <- dist(new_boston_scaled)
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970dist_man <- dist(new_boston_scaled, method = "manhattan")
summary(dist_man)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2662 8.4832 12.6090 13.5488 17.7568 48.8618library(RColorBrewer)
k_means <- kmeans(new_boston_scaled, centers = 3)
ggpairs(
as.data.frame(new_boston_scaled),
aes(color = as.factor(k_means$cluster)),
lower = list(continuous = wrap(
"smooth",
alpha = 0.3,
size = 0.5
)),
upper = list(continuous = wrap("cor", size = 2))
) +
scale_color_brewer(palette = "Dark2") +
scale_fill_brewer(palette = "Dark2") +
theme_bw()
twcss <- sapply(1:10, function(k){kmeans(new_boston_scaled, k)$tot.withinss})
qplot(x = 1:10, y = twcss, geom = "line") +
scale_x_continuous(breaks = c(1:10)) +
labs(x = "clusters") +
theme_bw()
new_k_means <- kmeans(new_boston_scaled, centers = 2)
ggpairs(
new_boston_scaled,
aes(color = as.factor(new_k_means$cluster)),
lower = list(continuous = wrap(
"smooth",
alpha = 0.3,
size = 0.5
)),
upper = list(continuous = wrap("cor", size = 2))
) +
scale_color_brewer(palette = "Dark2") +
scale_fill_brewer(palette = "Dark2") +
theme_bw()